St. Peter had an error on discipline regarding circumcision and St. Paul corrected him, yes this is true (Galatians 2:11-16). However, initially Protestants, but now even "Catholics" are using this reference as an excuse to arrogantly belittle the Pope with many kinds of derogatory remarks.

Let's examine the roles of St. Peter and St. Paul.

They are both Apostles of Jesus Christ, but Peter is the Rock where His Church was founded and he received the Keys of the Kingdom of Heaven (Matthew 16:18-19). Jesus prayed so that Peter's faith never fails and that he will confirm [teach] his brothers [us - and this, of course, also includes St. Paul] (Luke 22:31-32). Our Lord also told Peter, three times, to feed [teach and take care of] His sheep [us] (John 21:15-17). Was this privilege to the person of Peter only and not his successors ? No. Our Lord told the Apostles that He was going to be with them until the consummation of age (John 28:20), does this mean that Peter and all the Apostles were not be able to die and will be still living today and until the end of this world? No, Peter and the first Apostles died about 2,000 years ago, so this authority given from Christ to Peter is given to all Peter's successors because Jesus is with the Pope (the successor of Peter) and all the Bishops in communion with the Pope (the successor of the Apostles), generations after generations, until the consummation of age.

When the Apostle St. Paul corrected the First of the Apostles, St. Peter (Matthew 10:2), did he belittle him or was arrogant over him?

No, this is what St. Paul told St. Peter in front of everyone gathered at the place where they have met:

“If you, while you are a Jew, are living like the Gentiles and not the Jews, how is it that you compel the Gentiles to keep the customs of the Jews?”

Notice that St. Paul corrects St. Peter in a form of a question, not like he already has (or pretending to have) the infallible answer. Telling him what he believes in in the form of a question is actually a form of respect for Peter. St. Paul might have said it with strength, but still, it was said in a form of a question.

Does this seem to you like an excuse to say all kinds of derogative remarks towards our Vicar of Christ?

Yet, many persons who have not even called to be Bishops, arrogantly belittle our Pope, use this passage to spread their venom with all kinds of derogative remarks and like they themselves are infallible and have no question or possibility of error; but not even a Bishop can disrespect the Pope with derogative remarks.

Now, what does St. Peter, our first Pope, has to say about this situation?

[2 Peter]
{3:15} And let the longsuffering of our Lord be considered salvation, as also our most beloved brother Paul, according to the wisdom given to him, has written to you, {3:16} just as he also spoke in all of his epistles about these things. In these, there are certain things which are difficult to understand, which the unlearned and the unsteady distort, as they also do the other Scriptures, to their own destruction. {3:17} But since you, brothers, know these things beforehand, be cautious, lest by being drawn into the error of the foolish, you may fall away from your own steadfastness. {3:18} Yet truly, increase in grace and in the knowledge of our Lord and Savior Jesus Christ. To him be glory, both now and in the day of eternity. Amen.

Many are using these writings of St. Paul (as of all Scripture) in order to twist the meaning for their own foolish convenience. The use of these passages of the Galatians in order to have it (and even trump it) as an excuse to derogate, disrespect, belittle and say whatever they please against the Pope (even acting as if they themselves have more authority and are over the Pope), and also without giving any space that they themselves might be in error, is a clear indication that those who do that are nothing more than errors of the foolish and the unlearned to their own destruction as St. Peter himself teaches us, warn us, and clarify us regarding Sacred Scripture writings and interpretations.

At the very extreme case where you think that you are correct on a matter of discipline or opinions, and you think that the Pope is in error, then disagree but with respect.

Any valid Pope will never be a heretic or will lead us away from the path of salvation. How do we know ? Jesus Christ said so (Matthew 16:18 ) and He cannot err or lie because He is God who in turn is Truth Himself, that is His Nature and this will suffice.

[2 Peter 2]
{2:1} But there were also false prophets among the people, just as there will be among you lying teachers, who will introduce divisions of perdition, and they will deny him who bought them, the Lord, bringing upon themselves swift destruction.

"And they will deny him who bought them" Who brought them His Church? under what Rock? Peter and his successors. To deny the way His Church has been instituted is to deny Jesus Christ Himself.

Any valid Pope, whether Saint Peter himself or any of his successors, can err in a personal opinion, and can err in a prudential judgment or a decision on discipline, and can err, only to a limited extent, in a non-infallible teaching.

However, no Pope can teach heresy, neither as a personal opinion, nor as an apparent magisterial teaching, and no Pope can commit the sins of apostasy or heresy or schism, not even hidden in his heart and mind.

The teachings of the Pope are only infallible when they meet the criteria taught by the First Vatican Council.